A successful mathematical description of the renal processes requires an
understanding of the mechanisms through which these pressures take place.
Part of the present thesis addresses the hypothesis that increased coupling
between neighboring nephrons and increased strength of the tubuloglomerular
feedback process can explain the experimentally observed irregular oscillations
in the nephron pressures and flows. The hypothesis is put to test by
calculating Lyapunov exponents of a high level mechanism-based model of a
nephron and a similar model of two vascular coupled nephrons.
Synchronization between oscillating period-doubling systems is the topic
of the larger part of the study. Since synchronization is a fundamental phenomenon
in all sciences, it is treated from a general viewpoint by analyzing
one of the most simple dynamical systems, the R¨ossler system, both in an
externally forced version and in the form of two mutually coupled oscillators.
The bifurcational mechanism to resonant dynamics and chaotic phase
synchronization is described in detail. The transition from synchronized to
non-synchronized dynamics is known to take place at a dense set of saddlenode
bifurcations that run along the edge of the resonance tongue and appear
also to be related to the formation of multilayered tori and torus-doubling
bifurcations. A cyclic behavior of sub- and supercriticality of the period
doublings in the neighborhood of the contact between period doubling and
saddle-node bifurcations cause a set of torus bifurcations that take place at
a very small range of parameters.
In coupled R¨ossler systems, the same torus bifurcations take a more global
role. While a complete, but now folded, period-doubling cascade evolves,
a cascade of torus bifurcations emerge from all the period doublings and
run along side with three (due to the folding of the period doubling) sets
of saddle-node bifurcations at the edge of the tongue. Through homoclinic
bifurcations of tori with different periodicity, a second mechanism to phase
synchronization is found to occur.
Similar bifurcation structures are shown to exist in an externally forced
nephron model and in a model of two vascular coupled nephrons, underlining
that the discussed phenomena are of a common nature to forced and coupled
period-doubling systems.